Applying Cuckoo Search Algorithm to Solve …II. M ATHEMATICAL M ODEL FOR SOLVING FRDE For positive real number v, 01d dn v n, the order Caputo fractional derivative of the function

$Page 1: Applying Cuckoo Search Algorithm to Solve …II. M ATHEMATICAL M ODEL FOR SOLVING FRDE For positive real number v, 01d dn v n, the order Caputo fractional derivative of the function$
Abstract—In this paper, a swarm intelligence technique is

presented for solving the fractional differential equations (FDE)

based on the cubic spline function and cuckoo search algorithm.

In this technique, we approximate FDE with the cubic spline

approximation and use the cuckoo search algorithm as a tool for

the accurate and rapid solution. Based on this method, the given

problem is transformed into a problem for solving a nonlinear

equation system, and by solving this system, we obtain the

solution of FDE. Furthermore, special attention is given to the

error analysis of this method. The presented scheme is evaluated

on two initial value problems of FDE. The numerical simulation

results demonstrate that the proposed algorithm has higher

accuracy and is feasible and effective in solving initial value

problem of FDE.

Index Terms—fractional differential equation, cubic spline

approximation, cuckoo search algorithm

I. INTRODUCTION

ITH the development of science and technology, great

breakthroughs have been made in theoretical analysis

and numerical algorithm of fractional calculus. Analytical

solutions of fractional differential equations are usually

represented by some special functions, such as Green function,

Mittag-Leffler function, and so on. Up to now, the main

methods to solve the analytical solutions of fractional

differential equations include Fourier transform, Mellin

transform, Laplace transform, etc. However, it is an extremely

difficult task to find the analytical solution for general

fractional differential equation. Therefore, many researchers

tend to use numerical approaches to solve the fractional

differential equations in the recent thirty years, including

linear multi-step method[1-2], finite difference method[3-4],

Adomian decomposition method[5], homotopy perturbation

method[6-7], variational iteration method[8-9], artificial

neural network method[10], collocation method[11-12] and

Manuscript received March 14, 2019; revised September 16, 2019. This

work was supported by the Natural Science Foundation of Guangdong

Province under Grant No. 2017A030313280

Xinming Zhang (corresponding author) is with the College of Science of

Harbin Institute of Technology（Shenzhen）, Shenzhen, Guangdong,

518055 PR China(e-mail: [email protected]).

He Huang is with the Department of Probability and statistics of Harbin

Institute of Technology （Shenzhen）, Shenzhen, Guangdong,

518055 PR China(e-mail: [email protected]).

Xi Zhang is with the Department of Applied Mathematics of Harbin

Institute of Technology （Shenzhen）, Shenzhen, Guangdong,

518055 PR China(e-mail:[email protected]).

operation matrix method[13-14], the Legendre wavelet

method[15], Chebyshev wavelets method[16], the hybrid

Taylor series expansion with MHPM[17] , Stochastic

methods[18-20], and so on. However, there is still room for

investigation into numerical methods which can improve

results in terms of accuracy and reliability, with better

convenience, e.g. modern intelligent optimization algorithms.

These random search algorithms generally based on

biological intelligence or physical phenomena and are not

perfect in theory. However, from a practical viewpoint, such

algorithms usually do not require the continuity and convexity

of the objective function and constraints. Even if there is no

analytical expression, it is quite adaptable to the uncertain

data in calculation, which can overcome the limitations of

traditional methods to some extent.

In this paper, Cuckoo Search (CS) algorithm along with

cubic splined approximation is used, for the first time as per

our literature survey, to solve fractional differential equation.

The CS is a swarm intelligent algorithm developed by Yang

and Deb in 2009 that inspired from the nature[21]. Due to its

favorable efficiency, CS has been attracting considerable

attentions since it was born and has shown promising

superiority in many science and engineering fields, such as

inverse problems and shape optimization[22], phase

equilibrium and stability calculations[23], structural

optimization[24], hydraulic parameter estimation

problem[25], solution of nonlinear equation system[26],

multi-objective optimal power flow[27], hyperspectral image

classification[28], and so on. To the best of our knowledge,

the application of CS along with cubic splined approximation

(CS-CS) to solve fractional differential equation has not been

reported, and for the first time, this topic is investigated in the

literature. The aim of our study is to identify the relative

strengths of the proposed algorithm for the solution of

fractional differential equation. This study shows that CS-CS

algorithm offers a reliable performance for solving these

differential equation of fractional order.

The rest of this paper is organized as follows: In Section 2

some basic definitions and the proposed numerical algorithm

(CS-CS) are given. Section 3 shows the error analysis of the

presented method. In Section 4, several numerical

experiments are conducted to verify the feasibility and

effectiveness of the proposed method for solving fractional

differential equations, and finally, the conclusions are derived

in Section 5.

Xinming Zhang, He Huang, and Xi Zhang

Applying Cuckoo Search Algorithm to Solve

Fractional Differential Equation Based on Cubic

Spline Function

W

IAENG International Journal of Applied Mathematics, 50:1, IJAM_50_1_05

Volume 50, Issue 1: March 2020

______________________________________________________________________________________

II. MATHEMATICAL MODEL FOR SOLVING FRDE

For positive real number v , 0 1n v n , the order

v Caputo fractional derivative of the function ( )f t defined

on the interval [0, ]T is

10

1, 0 1 ,

, .

nt

v n

C v

a tn

n

f dif n v n

n v tD f t

df t if v n N

dt

(1)

The generic non-linear quadratic fractional differential

equations solved in this article can be written as

2( )

( ) ( ) ( ) ( ) ( ), 0

v

v

d y tp t q t y t r t y t t T

dt (2)

with initial condition given as

(0) , 0,1,2,..., 1k

kk

dy c k n

dt , (3)

where v is the order which satisfies 0v , v , n v .

( )y t is the solution of the fractional differential equation.

( ), ( ), ( )p t q t r t are known functions, and ,k

T c are known

parameters.

A. Cubic spline approximation

For the initial value problem of fractional differential

equation (2) and (3), in this sub-section, we discretize

fractional differential equations into nonlinear algebraic

equations by cubic spline function.

Taking 1m nodes on the interval [0, ]T and dividing the

interval into m subintervals 1 2 2 3 m 1[ , ],[ , ],...,[ , ]mt t t t t t

.Using

a cubic spline function on each subinterval, that is 3 2

1( ) , , 1,2,..., .i i i i i i iy t a t b t c t d t t t i m (4)

Since the second derivative of the cubic spline is

continuous, we have

1 11

ˆ ˆ( ) ( ) , 1,2,..., 1i i

i it t t ty t y t i m

1 1

(1) (1)

1ˆ ˆ( ) ( ) , 1,2,..., 1

i ii it t t t

y t y t i m

1 1

(2) (2)

1ˆ ˆ( ) ( ) , 1,2,..., 1

i ii it t t t

y t y t i m

Since the above equations satisfy the initial condition

0ˆ(0)y y , we can get

3 2

0 0tat bt ct d y , that is

0d y .

Then take l small nodes on each subinterval, and let the

function value of each small node approximate the fractional

differential equation. So there is 2 ( )( ) ( ) ( ) ( ) ( ) ( ) , 1,2,..., .v

i i ip t q t y t r t y t y t i m l (5)

Thus, the problem is transformed into the following

nonlinear algebraic equations with undetermined coefficients.

1) Case 1. 0 1v ,

1 1

1 1

1 1

1

1

1

1 0

2

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ),

1,2,...,

i i

i i

i i

i t t i t t

i t t i t t

i t t i t t

v

i

y t y t i m

y t y t i m

y t y t i m

d y

y t p t q t y t r t y t

i m l

(6)

where

t( ) 2

0

2

0

0 0

2 1

0

1 1

0

1ˆ ( ) ( ) (3 2 )d

(1 )

3( ) d

(1 )

2( ) d ( ) d

(1 ) (1 )

3d( )

(1 ) (1 )

2d( )

(1 ) (1 ) (1 ) (1 )

6

(1 ) (1

v v

i i i i

tvi

t tv vi i

tvi

tv vi i

i

y t t a b cv

at

v

c bt t

v v

at

v v

c bt t

v v v v

a

v

1

0

1 2

3

1 2

( ) d)

2

(1 ) (1 ) (2 )(1 ) (1 )

6

(3 )(2 )(1 ) (1 )

2.

(1 ) (1 ) (2 )(1 ) (1 )

tv

v vi i

vi

v vi i

tv

c bt t

v v v v v

at

v v v v

c bt t

v v v v v

(7)

2) Case 2. 1 2v ,

1 1

1 1

1 1

1

1

1

1 0

1 0

2

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ),

1,2,..., .

i i

i i

i i

i t t i t t

i t t i t t

i t t i t t

v

i

y t y t i m

y t y t i m

y t y t i m

d y

c y


i m l

(8)

where

1t( )

0

1 1

0 0

2

22

00

2 3

1ˆ ( ) ( ) (6 2 )d

(2 )

2 6( ) d ( ) d

(2 ) (2 )

2

(2 ) (2 )

6( ) ( ) d

(2 ) (2 )

2 6.

(2 ) (2 ) (3 )(2 ) (2 )

vv

i i i

v vt ti i

vi

vtv ti

v vi i

y t t a bv

b at t

v v

bt

v v

at t

v v

b at t

v v v v v

(9)



______________________________________________________________________________________

3) Case 3. 2 3v ,

1 1

1 1

1 1

1

1

1

1 0

1 0

1 0

2

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

2

ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ),

1,2,..., .

i i

i i

i i

i t t i t t

i t t i t t

i t t i t t

v

i

y t y t i m

y t y t i m

y t y t i m

d y

c y

b y


i m l

(10)

where 2

( )

0

3

0

3

3

1ˆ ( ) ( ) 6 d

(3 )

6 1( )

(3 ) 3

6 10

(3 ) 3

6 1.

(3 ) 3

vtv

i i

v ti

vi

vi

y t t av

at

v v

at

v v

at

v v

(11)

To sum up, transforming the initial value problem of

fractional differential equation into a nonlinear equation

model involves three steps. First, we divide the interval

[0, ]T into m subintervals , 1 2 2 3 m 1[ , ],[ , ],...,[ , ]mt t t t t t

, and

express the unknown function with cubic spline

function in each interval 3 2ˆ ( )i i i i iy t a t b t c t d ,

1i it t t 1, 2,...,i m .Seco

nd, based on the fact that the second derivative of cubic spline

function is continuous, we transform the fractional differential

equation into a model of nonlinear equations. Third, for the

part ( )ˆ ( )v

iy t which is difficult to solve in the nonlinear

equations model, we discuss it in several cases.

B. Implementation of cuckoo search algorithm

After transforming the fractional differential equation into

a nonlinear equations model, common traditional algorithms

for solving nonlinear algebraic equations include newton

method, conjugate gradient method, least squares method, etc.

However, it has been pointed out that these methods require

extremely high in the selection of initial points. In recent years,

several modern intelligent algorithms have been developed to

calculate nonlinear equations, such as Genetic algorithm,

particle swarm algorithm, ant colony algorithm, etc. These

algorithms not only overcome the problem of selecting the

initial points existing in traditional algorithms, but also have

strong global optimization ability to some extent. In this

section, we will use the cuckoo search algorithm to solve the

transformed nonlinear equations. In order to better implement

the cuckoo search algorithm, three ideal states are assumed:

(1) Each cuckoo produces only one egg at a time, and

randomly chooses a nest to place it;

(2) In the process of searching bird's nest, we calculate the

optimal nest position and save it to the next generation;

(3) The probability of host finding cuckoo’s eggs and

abandoning them is , [0,1]a ap p .

Thus, the update formula of cuckoo search can be

expressed as

( 1) ( ) ( ), 1, 2,3,...,t t

i ix x L i n , (12)

where ( )t

ix indicates the position of the i th bird nest in the

t th iteration; represents the step size, usually take 1 ;

L obeys Levy distribution

( )

1( ) 0.01 ( ), 0 2,t t

j i

uL x x

v

(13)

,u v obeys normal distribution, 2 2~ 0, , ~ 0,u vu N v N ,

1

1 2

1 sin 2

1 2 2

1.

u

v

(14)

In this way, we can refer to the cuckoo search for the

optimal nest and hatching process to implement the cuckoo

search algorithm. The specific steps are as follows:

Step 1: Define and initialize the objective

function1 2( ), ( , , ..., )T

df X X x x x , d is the dimension of

bird's nest, and randomly generate n initial nest position

( 1, 2,..., )iX i n . Initialize the rejection probabilityap .

Step 2: Calculate the value of objective function of each

bird nest position, and select the bird nest with the optimal

value.

Step 3: Preserve the optimal nest location of the previous

generation, and update the nest location with Levy flight (13).

Step 4: Compare the current value of position function with

the previous optimal value. If better, update the value of the

current objective function, otherwise, retain the optimal value

of the previous generation.

Step 5: After updating the position, generate the random

number [0,1]r . If ar p , renew

( 1)t

ix and compare the

new nest, then calculate the global position*

tpb .

Step 6: Determine whether *( )tf pb meets the maximum

iterations or minimum error requirement, and if so, the output *( )tf pb is the global optimal solution gb .Otherwise, return

to step 2.

III. ERROR ANALYSIS

In this section, we will analyze the error of approximating

the solution function of the fractional differential equation

with cubic spline function, and derive the convergence order.

For general nonlinear quadratic fractional differential

equation

2( )( ) ( ) ( ) ( ) ( ), [0, ] 0,

v

v

d y tp t q t y t r t y t t T v

dt , (15)

(0) , 0,1, 2, ..., 1,k

kk

dy C k n n v

dt , (16)

Taking 1m nodes on the interval [0, ]T and dividing the

interval into m subintervals, 1 2 2 3 m 1[ , ],[ , ],...,[ , ]mt t t t t t

.

Using a cubic spline function on each subinterval, one gets

3 2

1( ) , ,

1,2,..., .

i i i i i iiy t a t b t c t d t t t

i m

(17)



______________________________________________________________________________________

and

2ˆ ˆ( ) 3 2 , ( )

ˆ6 2 , ( ) 6

i i i i i

i i i i

y t a t b t c y t

a t b y t a

(18)

Using Taylor expansion on the interval 1[ , ]i it t , we have

3 2

3 2

2

2 3 3

2

3 3

ˆ ( )

(3 2 )( )

(6 2 ) 6( ) ( ) ( )

2! 3!

ˆ ( )ˆ ˆ( ) ( )( ) ( )

2!

ˆ ( )( ) ( ),

3!

i i i i i

i i i i i i i

i i i i i i

i i i ii i

i ii i i i i i

i ii

y t a t b t c t d

a t b t c t d

a t b t c t t

a t b at t t t o h

y ty t y t t t t t

y tt t o h

(19)

2

2

2 2

ˆ ( ) 3 2

3 2 (6 2 )( )

6( ) ( ),

2!

i i i i

i i i i i i i i i

ii

y t a t b t c

a t b t c a t b t t

at t o h

(20)

ˆ ( ) 6 2

6 2 6 ( ) ( ).

i i i

i i i i i

y t a t b

a t b a t t o h

(21)

The real analytic solution ( )iy t of (15) and (16) can be

expanded to

2

3

( )( ) ( ) ( )( ) ( )

2!

( ) ( )( ) ... ( ) ( )

3! !

i ii i i i i i i

nn ni i i i

i i

y ty t y t y t t t t t

y t y tt t t t o h

n

2

( )1 1

( )( ) ( ) ( )( ) ( )

2!

( )... ( ) ( )

( 1)!

i ii i i i i i i

nn ni i

i

y ty t y t y t t t t t

y tt t o h

n

( )2 2

( ) ( ) ( )( )

( )... ( ) ( )

( 2)!

i i i i i i

nn ni i

i

y t y t y t t t

y tt t o h

n

Then, we get

2 3 3

(4) ( )4

( ) ( )

( ) ( ) ( ( ) ( ))( )

( ) ( ) ( ) ( )( )( ) ( )( ) ( )

2! 2! 3! 3!

( ) ( )( ) ... ( ) ( )

4! !

i i

i i i i i i i i i

i i i i i i i ii i

nn ni i i i

i i

y t y t

y t y t y t y t t t

y t y t y t y tt t t t o h

y t y tt t t t o h

n

2 2

(4) ( )3 1 1

( ) ( )

( ) ( ) ( ( ) ( ))( )

( ) ( )( )( ) ( )

2! 2!

( ) ( )( ) ... ( ) ( )

3! ( 1)!

i i

i i i i i i i i i

i i i ii

nn ni i i i

i i

y t y t

y t y t y t y t t t

y t y tt t o h

y t y tt t t t o h

n

(4) (5)2 3

( )2 2

( ) ( )

( ) ( ) ( ( ) ( ))( ) ( )

( ) ( )( ) ( )

2! 3!

( )... ( ) ( )

( 2)!

i i

i i i i i i i i i

i i i ii i

nn ni i

i

y t y t

y t y t y t y t t t o h

y t y tt t t t

y tt t o h

n

According to (10), fractional differential equation (15) and

(16) can be reduced to the following equations

1 1

1 1

1 1

1

1

1

1 0 1

1 0 2

1 0 3

2

( ) ( ) , 1,2,..., 1

( ) ( ) , 1,2,..., 1

( ) ( ) , 1,2,..., 1

2

( ) ( ) ( ) ( ) ( ) ( ),

1,2,...,

i i

i i

i i

i t t i t t

i t t i t t

i t t i t t

v

i

y t y t i m

y t y t i m

y t y t i m

d y C

c y C

b y C


i m l

(22)

Theorem 1

If the analytical solution ( )y t of the problem (15) and (16)

has n order continuous derivative on the interval [0, ]T , then

the local truncation error of the approximate solution function

ˆ ( ), 1, 2,...iy t i m used to simulate the real analytic solution

( )y t is 3( )o h .

Proof:

On the interval 1 2[ , ]t t , where

1 0t ,

1 1

1 1 1 1 1 1 1 1 1

2 31 1 11 11

(4) ( )3 41 1 1

1 1

3 11 1 2 1

( ) ( )

( ) ( ) ( ( ) ( ))( )

( ) ( ) ( )( )( )( ) ( )( )

2! 2! 3! 3!

( ) ( )( ) ( ) ... ( ) ( )

4! !

2( ) ( )

2! 2!

i i ii

nn ni

y t y t

y t y t y t y t t t

y t y t y ty tt t t t

y t y to h t t t t o h

n

C bC d C c t

2 31 1

(4) ( )3 41 1

3 31 1

(4) ( )41 1

31 1

(0) 6( )

3! 3!

(0) (0)( ) ... ( )

4! !

(0) 6= ( ) ( )

3! 3!

(0) (0)... ( )

4! !

(0) 61 ( )

3! 3!

nn n

nn n

y at t

y yo h t t o h

n

y at o h

y yt t o h

n

y ao h



______________________________________________________________________________________

1 1

1 1 1 1 1 1 1 1 1

2 21 1 1 11

(4) ( )3 1 11 1 1 1

1 1

21 12 1 3 1

2

( ) ( )

( ) ( ) ( ( ) ( ))( )

( ) ( )( )( ) ( )

2! 2!

( ) ( )( ) ... ( ) ( )

3! ( 1)!

(0) 6( 2 ) ( )

2! 2!

( )

nn n

y t y t

y t y t y t y t t t

y t y tt t o h

y t y tt t t t o h

n

y aC c C b t t

o h

(4) ( )

3 1 11 1

2 21 1

(4) ( )3 1 11 1

21 1

(0) (0)... ( )

3! ( 1)!

(0) 6( ) ( )

2! 2!

(0) (0)... ( )

3! ( 1)!

(0) 61 ( )

2! 2!

nn n

nn n

y yt t o h

n

y at o h

y yt t o h

n

y ao h

1 1

1 1 1 1 1 1 1 1 1

(4) (5)2 31 1 1 1

1

( )2 21 1

1

2 1 1 1 1

(4) (5)21 1

( ) ( )

( ) ( ) ( ( ) ( ))( ) ( )

( ) ( )( ) ( )

2! 3!

( )... ( ) ( )

( 2)!

2 ( ( ) 6 ) ( )

(0) (0)

2! 3!

i

nn n

y t y t

y t y t y t y t t t o h

y t y tt t t t

y tt t o h

n

C b y t a t o h

y yt

( )

3 2 21

1 1

(0)... ( )

( 2)!

(0) 6 1 ( )

nn ny

t t o hn

y a o h

On the interval 2 3[ , ]t t ,

2 2

2 2 2 2 2 2 2 2 2

2 3 32 2 2 2 2 2 2 22 2

(4) ( )42 2 2 2

2 2

1 2 1 2 1 2

ˆ( ) ( )

ˆ ˆ( ) ( ) ( ( ) ( ))( )

ˆ ˆ( ) ( ) ( ) ( )( )( ) ( )( ) ( )

2! 2! 3! 3!

ˆ ˆ( ) ( )( ) ... ( ) ( )

4! !

ˆ ˆ( ) ( ) ( )

nn n

y t y t

y t y t y t y t t t

y t y t y t y tt t t t o h

y t y tt t t t o h

n

y t y t y t

1 2

2 3 31 2 1 2 2 2 2 22

(4) ( )42 2 2 2

2 2

( ) ( )

ˆ ˆ( ) ( ) ( ) ( )( ) ( )( ) ( )

2! 2! 3! 3!

ˆ ˆ( ) ( )( ) ... ( ) ( )

4! !

nn n

y t o h

y t y t y t y to h t t o h

y t y tt t t t o h

n

3 21 1 1 1

2 32 2 2 21 1

(4) ( )42 2 2 2

2 2

1 1 1

(0) 6 (0) 61 ( ) 1 ( ) ( )

3! 3! 2! 2!

ˆ( ) ( )1(0) 6 1 ( ) ( ) 1 ( )

2! 3! 3!

ˆ ˆ( ) ( )( ) ... ( ) ( )

4! !

(0) 6 (0) 61

3! 3! 2!

nn n

y a y ao h o h o h

y t y ty a o h o h o h

y t y tt t t t o h

n

y a y

1

32 2 2 21 1

12!

ˆ( ) ( )1(0) 6 1 1 ( ).

2! 3! 3!

a

y t y ty a o h

In the same way, on each interval

1 2 2 3 1[ , ],[ , ],...,[ , ]m mt t t t t t , we have

3( ) ( ) ( )i i iy t y t C o h

where iC is an appropriate positive constant, 1, 2,...,i m .

IV. NUMERICAL EXAMPLES

In order to verify the feasibility and effectiveness of the

new proposed method for solving fractional differential

equations. In this section, we will present two numerical

experiments based on the previous discussion.

A. Example 1

Consider fractional differential equation

2 2( ) 2( ), 0, (0) 0, 0 1.

(3 )

vv

v

d y tt t y t t y v

vdt

(23)

The exact solution of this equation is 2( ) .y t t

According to (6), (23) can be transformed into the

following equations:

1 1

1 1

1 1

1

1 1

1

(2) (2)

1

1

2 2 1

2

( ) ( ) , 1,2,... 1

( ) ( ) , 1,2,... 1

( ) ( ) , 1,2,... 1

0

20 ( )

(3 ) (1 ) (1 )

2 6

(2 )(1 ) (1 ) (3

i i

i i

i i

i it t t t

i it t t t

i it t t t

v vii

vi i

y t y t i m

y t y t i m

y t y t i m

d

ct t y t t

v v v

b at

v v v

（）（）

3 ,)(2 )(1 ) (1 )

1,2,...,

vtv v v v

i m l

For convenience, we first select

0.5, 1, 20, 1v m l T . At this time, the error tolerance

can reach 1510Tol , which takes 12.313966 seconds.

Moreover, in order to increase the credibility of the numerical

simulation, the results are averaged by considering 30

different executions. The comparison results are shown in

Fig.1. As we can see that the solution obtained by CS-CS is

almost close to the real analytical solution. Table I lists the

numerical solution and the error comparison of these methods.

From Table I, we can find that the accuracy of the cuckoo

search algorithm based on cubic spline approximation

(CS-CS) is much higher than the particle swarm optimization

based on artificial neural network (PSO-ANN) and

Grunwald Letnikov classical numerical method[19].



______________________________________________________________________________________

TABLE I

PSO-ANN, GL, CS-CS COMPARISON TABLE AT t = [0,1]

t Exact

solution

Solution of

GL

Solution of

PSO-ANN

Solution of

CS-CS

Error by

GL

Error by

PSO-ANN

Error by

CS-CS

0.0 0.00000 0.00000 5.45e-6 0.00000000000000 0 5.45e-6 5.2e-18

0.2 0.04000 0.04007 0.04045 0.04000000000000 7e-5 4.52e-4 0

0.3 0.09000 0.09011 0.09072 0.09000000000000 1.1e-4 7.18e-4 0

0.4 0.16000 0.16013 0.16045 0.16000000000000 1.3e-4 4.51e-4 0

0.5 0.25000 0.25016 0.24958 0.25000000000000 1.6e-4 4.25e-4 0

0.6 0.36000 0.36019 0.35827 0.36000000000000 1.9e-4 1.73e-3 -5.551e-17

0.7 0.49000 0.49021 0.48693 0.49000000000000 2.1e-4 3.07e-3 -5.551e-17

0.8 0.64000 0.64023 0.63619 0.64000000000000 2.3e-4 3.81e-3 0

0.9 0.81000 0.81026 0.80688 0.81000000000000 2.6e-4 3.11e-3 0

1.0 1.00000 1.00028 1.00004 1.00000000000000 2.8e-4 4.40e-5 0

Fig. 1. Comparison of PSO-ANN, GL, CS-CS at t = [0, 1] Fig. 2. Simulation diagram of by CS-CS at t = [0, 2]

Fig. 3. Simulation diagram of by CS-CS at t = [0, 10] Fig.4. Simulation diagram of by CS-CS at t = [0, 100]

CS-CS

Exact

GL

PSO-ANN



______________________________________________________________________________________

TABLE II

COMPARISON OF RESULTS FOR THE SOLUTION

OF EXAMPLE 1 (v = 0.25, t = [0, 1])

t Exact solution Error by EOC Error by FWCW Mean error by CS-CS

0.09375 0.0087890625 1.15137021309e-5 0.43017672757 e-13 1.243218491116710e-17

0.18750 0.0351562500 1.34008681836e-5 0.53908266740 e-13 1.989149585786740e-17

0.28125 0.0791015625 1.45485346092e-5 0.24466539905 e-13 1.850371707708590e-17

0.37500 0.1406250000 1.53757162245e-5 0.03941291737 e-13 1.757853122323160e-17

0.46875 0.2197265625 1.60223410662e-5 0.64837024638 e-13 2.312964634635740e-17

0.56250 0.3164062500 1.65528139359e-5 0.54012350148 e-13 2.035408878479450e-17

0.65625 0.4306640625 1.70022170524e-5 0.09825473768 e-13 3.515706244646330e-17

0.75000 0.5625000000 1.73918021056e-5 0.17985612999 e-13 1.480297366166880e-17

0.84375 0.7119406250 1.77354280767e-5 0.31863400807 e-13 3.330669073875470e-17

0.93750 0.8789062500 1.80426383865e-5 2.01283434365 e-13 4.810966440042350e-17

TABLE III

COMPARISON OF RESULTS FOR THE SOLUTION

OF EXAMPLE 1 (v = 0.75, t = [0, 1])

t Exact solution Error by EOC Error by FWCW Mean error by CS-CS

0.09375 0.0087890625 0.201409844130e-3 0.347308987125e-13 1.295260195396020e-17

0.18750 0.0351562500 0.312234854242e-3 0.022967738822e-13 2.012279232133100e-17

0.28125 0.0791015625 0.397141377633e-3 0.198174809896e-13 2.359223927328460e-17

0.37500 0.1406250000 0.466219134874e-3 0.293931545769e-13 2.683038976177460e-17

0.46875 0.2197265625 0.524234122143e-3 0.221489493413e-13 2.220446049250310e-17

0.56250 0.3164062500 0.573965121404e-3 0.194289029309e-13 2.775557561562890e-17

0.65625 0.4306640625 0.617222689041e-3 0.170974345793e-13 2.775557561562890e-17

0.75000 0.5625000000 0.655271385754e-3 0.091038288019e-13 2.220446049250310e-17

0.84375 0.7119406250 0.689037602412e-3 0.119904086659e-13 4.810966440042350e-17

0.93750 0.8789062500 0.719224088406e-3 0.275335310107e-13 6.661338147750940e-17

Fig.5. Error comparison of EOC,FWCW,CS-CS at v = 0.25, t = [0, 1] Fig.6. Error comparison of EOC,FWCW,CS-CS at v = 0.75, t = [0, 1]

In addition, CS-CS not only has high accuracy at interval

[0,1]t , but also maintains high accuracy and stability

when the range of T becomes larger. Fig. 2, 3, 4 are the

simulated diagram of numerical solutions obtained by

EOC

FWCW

CS-CS

EOC

FWCW

CS-CS



______________________________________________________________________________________

CS-CS at intervals [0, 2]t , [0,10]t and [0,100]t ,

respectively. These three cases take 15.068018 seconds,

17.216479 seconds and 20.749634 seconds, respectively.

Even for the case [0,100]t , the accuracy of the

numerical solution can be as high as 1310

, which shows

that the algorithm still has high accuracy and stability when

the T value keeps increasing.

In order to show the superiority of CS-CS in further, the

value of the fractional order derivative ν is taken as 0.25

and 0.75. The corresponding results are summarized in

Table II, Table III and Fig.5, Fig.6. It also contains

reported results with FWCM[29] and EOC[30]. It can be

inferred that our algorithm provides an approximate

solution to the fractional differential equation more

effectively.

B. Example 2

Consider the fractional differential equation 0.8

1.8

0.8

2 2.8

d ( ) 14( ) (1 ) ( )

(3.8)d

5 5(1 )

2 (3.8)

y ty t t y t t

t

t t t

(24)

The initial condition is

(0) 0y

The exact solution of this equation is

2.85( )

(3.8)y t t


following equations

1 1

1 1

1 1

1

1 1

1

(2) (2)

1

1

1.8 2

2.8

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

0

14 5ˆ ˆ0 ( ) (1 ) ( )

(3.8) 2

5(1 )

(3.8) (

i i

i i

i i

i it t t t

i it t t t

i it t t t

i

y t y t i m

y t y t i m

y t y t i m

d

y t t y t t t

ct t

（）（）

1

2 3

1 ) (1 )

2 6,

(2 )(1 ) (1 ) (3 )(2 )(1 ) (1 )

1,2,..., .

v

v vi i

tv v

b at t

v v v v v v v

i m l

In the following, we will use cuckoo search algorithm to

calculate the above equations and discuss the influence of

the parameter l .

(1)Choose 2, 10, 1, 1.1227m l T Tol . It takes

16.439011 seconds. The coefficient solutions are:

1 2

1 2

1 2

1 2

0.925561212447776 0.738891012901748

0.179615628191602 0.459623589255583

0.015345206321721 0.155256770376138

0.000000008744638 0.023283177842368,

a a

b band

c c

d d

We get the approximate solution of the equation:

3 2

1

3 2

2

ˆ( )

( ) 0.925561212447776 0.179615628191602

0.015345206321721 0.000000008744638 0 0.5

( ) 0.738891012901748 0.459623589255583

0.155256770376138 0.023283177842368 0

y t

y t t t

t t

y t t t

t

.5 1.t

(2) Choose 2, 20, 1, 1.1714m l T Tol . It takes


1 2

1 2

1 2

1 2

0.919498494305771 0.750548643123485

0.183727300560656 0.437107118636707

0.015943443439072 0.142555742170239

0.000000014508539 0.021816521494590,

a a

b band

c c

d d

The approximate solution of the equation is:

3 2

1

3 2

2

ˆ( )

( ) 0.919498494305771 0.183727300560656

0.015943443439072 0.000000014508539 0 0.5

( ) 0.750548643123485 0.437107118636707

0.142555742170239 0.021816521494590 0

y t

y t t t

t t

y t t t

t

.5 1.t

(3) Choose 2, 30, 1, 1.2156m l T Tol . It takes


1 2

1 2

1 2

1 2

0.925913752517327 0.754732405733558

0.178866218377388 0.427150121741720

0.015223271257590 0.134842553318408

0.000000025245317 0.019562451330709,

a a

b band

c c

d d

This yields:

3 2

1

3 2

2

ˆ( )

( ) 0.925913752517327 0.178866218377388

0.015223271257590 0.000000025245317 0 0.5

( ) 0.754732405733558 0.427150121741720

0.134842553318408 0.019562451330709 0

y t

y t t t

t t

y t t t

t

.5 1.t

Table IV lists the error comparison between the

numerical solutions obtained by CS-CS and the analytical

solution when 10, 20,l l and 30l . The Mean square

error(MSE) of three cases are MSE_l10=2.2772e-5,

MSE_l20=1.7102e-5, MSE_l30=2.1554e-5, respectively.

We can find that the error is relatively small at around 510

for 20l .

Fig.7 shows the numerical solutions obtained with

CS-CS ( 20l ) and difference method(DM) in [31]. From

Fig.7, we can observe that the solution obtained by CS-CS

is closer to analytical solution than DM. In addition, Table

V lists the error comparison between the exact solutions

and numerical solutions by CS-CS and DM methods at

several points. It is shown that the numerical solution

accuracy of CS-CS is around4 510 ~ 10

, which is more

close to the real analytical solution than DM method. Table

VI lists the maximum error, the minimum error and the

mean error of CS-CS for 30 experiments.



______________________________________________________________________________________

Fig.7. Comparison of CS-CS,DM at t=[0,1],v=0.8

C. Example 3

Consider fractional differential equation

2 2d ( ) ( 1)( ) ,

d ( 1)

vp v p

v

y t pt y t t

t p v

(25)

The initial condition is

(0) 0y for 0 1v

(0) (0) 0y y for 1 2v

The exact solution of this equation is

( ) py t t

In the following, we will discuss the solution of (25) in

different cases.

Case 1. 3, 0.5p v


following equations

1 1

1 1

1 1

1

(1) (1)

1

(2) (2)

1

1

2 2 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

ˆ ˆ( ) ( ) , 1,2,..., 1

0

(4)0 ( )

(4 ) (1 ) (1 )

2

(2 )(1 ) (

i i

i i

i i

i t t i t t

i t t i t t

i t t i t t

p v p vii

i

y t y t i m

y t y t i m

y t y t i m

d

ct y t t t

v v v

b

v v

2 36,

1 ) (3 )(2 )(1 ) (1 )

1,2,..., .

v viat t

v v v v v

i m l

The parameter is chosen as

1 1, 10, 0.4182m T l Tol , .Using cuckoo algorithm

to calculate the above equations, and it takes 4.8807

seconds. The simulation results are shown in Fig.8 and

Table VII. As we can see that the numerical solution agree

well with the analytical solution and the mean error of

CS-CS can reach -710 .

2) Case 2. 3, 1.5p v


following equations

1 1

1 1

1 1

1

(1) (1)

1

(2) (2)

1

1

1

2

3

ˆ ˆ( ) ( ) , 1,2, , 1

ˆ ˆ( ) ( ) , 1,2, , 1

ˆ ˆ( ) ( ) , 1,2, , 1

0

0

(4) 20

(4 ) (2 ) (2 )

6ˆ

(3 )(2 ) (1 )

i i

i i

i i

i t t i t t

i t t i t t

i t t i t t

p v vi

vii

y t y t i m

y t y t i m

y t y t i m

d

c

bt t

v v v

at y

v v v

2 2( ) ,

1,2, , 1.

pt t

i m

Similarly, the parameter is chosen

as 151, 1 10, 10m T l Tol , . It takes 10.7952 seconds

for cuckoo algorithm to calculate the above equations.

Fig.9 and Table VIII give the simulation results. We can

conclude that the numerical solution are found in well

agreement with the analytical solution and the mean error

of CS-CS can reach-1610 .

V. CONCLUSION

In this paper, we propose a new solution scheme for the

initial value problem of fractional differential equations,

which is solved by cuckoo search algorithm based on cubic

spline (CS-CS). A cubic spline function was introduced to

transform the fractional differential equations into

nonlinear equations and the Cuckoo search algorithm was

applied to solve the nonlinear equation system.

Furthermore, we derive the convergence order which

proves the theoretical feasibility of the proposed method.

By using CS-CS algorithm to solve specific examples, we

find that the new method has the characteristics of high

precision and fast convergence speed. Therefore, the

cuckoo algorithm based on cubic spline (CS-CS) presented

in this paper is feasible and effective in solving the initial

value problem of fractional differential equations.

CS-CS

DM

Exact



______________________________________________________________________________________

Fig. 8. Numerical results of CS-CS for p = 3, v = 0.5 Fig. 9. Numerical results of CS-CS for p = 3, v = 1.5

TABLE IV

THE ERROR COMPARISON OF CS-CS (t=[0,1] , v = 0.8)

t Exact Solution Solution of CS-CS Error by CS-CS

l=10 l=20 l =30 l=10 l=20 l =30

0 0 0.000000008744

638

-0.000000014508

539

-0.000000025245

317

0.0000000087446

38

0.0000000145085

39

0.0000000252453

17 0.1 0.001688149100

350

0.001187205606

829

0.0011624126474

66

0.0011922235652

15

0.0005009434935

20

0.0005257364528

84

0.0004959255351

34

0.2 0.011756953201

898

0.010694555759

719

0.0115163767805

19

0.0115172792583

99

0.0002368708943

59

0.0002405764213

80

0.0002396739434

99

0.3 0.036588981023

069

0.036552006121

455

0.0365788688564

54

0.0365306243493

39

0.0000369749016

14

0.0000101121666

15

0.0000583566737

30

0.4 0.081880177860

470

0.081836344323

263

0.0818668798411

06

0.0817877413531

380

0.0000438335372

07

0.0000132980193

64

0.0000924365073

32

0.5 0.152942017148

660

0.152926464187

650

0.1528974007003

10

0.1528441127849

01

0.0000155529610

10

0.0000446164483

49

0.0000979043637

58

0.6 0.254820464320

514

0.255375732989

301

0.2551874223999

00

0.2552552211597

32

0.0005552686687

87

0.0003669580793

86

0.0004347568392

18

0.7 0.392360531068

552

0.394737518002

905

0.3942539359057

11

0.3945765489927

34

0.0023769869343

53

0.0018934048371

59

0.0022160179241

81

0.8 0.570246679673

755

0.576565186503

147

0.5756139321835

78

0.5763635787990

11

0.0063185068293

93

0.0053672525098

23

0.0061168991252

57

0.9 0.793030385781

202

0.806412105764

715

0.8047844021993

34

0.8061717930936

68

0.0134730381732

07

0.0117540164181

33

0.0131414073124

67

TABLE V

THE ERROR COMPARISON OF CS-CS, DM ( t=[0,1] , v = 0.8, l=20)

t Exact Solution Mean solution of CS-CS Solution of DM Mean error by CS-CS Error by DM

0 0 0.000000000140022

0.00000000 9.013090719625970e-09

0

0.1 0.001688149100350 0.001166142476867

0.00194987 5.220066234831710e-04

0.00026172

0.2 0.011756953201898 0.011516154287996

0.01284495 2.407989139025410e-04

0.00108709

0.3 0.036588981023069 0.036571078568839

0.03926037 1.790245423084970e-05

0.00267139

0.4 0.081880177860470 0.081851958314823

0.08701887 2.821954564697870e-05

0.00513869

0.5 0.152942017148660 0.152879836521379 0.16142110 6.218062728019690e-05 0.00847908

0.6 0.254820464320514 0.255175756183935 0.26732269 3.552918634213640e-04 0.01250222

0.7 0.392360531068552 0.394260760297919 0.40916953 1.900229229367280e-03 0.01680899

0.8 0.570246679673755 0.575655891858762 0.59101864 5.409212185006990e-03 0.02077195

0.9 0.793030385781202 0.804882193861890 0.81655453 1.185180808068810e-02 0.02395547



______________________________________________________________________________________

TABLE VI

THE ERROR OF CS-CS (t=[0,1],v=0.8,l=20)

t Min error by CS-CS Max error by CS-CS Mean error by CS-CS

0 1.702279956823930e-09 3.888834435686740e-10 9.013090719625970e-09

0.1 5.265364140468620e-04 5.191041327268500e-04 5.220066234831710e-04

0.2 2.437314167992880e-04 2.409312625101370e-04 2.407989139025410e-04

0.3 1.818830810562670e-05 2.247311334883930e-05 1.790245423084970e-05

0.4 2.988866827845220e-05 3.409837630968800e-05 2.821954564697870e-05

0.5 7.434221558075270e-05 6.170388040974140e-05 6.218062728019690e-05

0.6 3.184493334056350e-04 3.743215470149440e-04 3.552918634213640e-04

0.7 1.819438002456540e-03 1.954542818986270e-03 1.900229229367280e-03

0.8 5.260125226887680e-03 5.520074260065730e-03 5.409212185006990e-03

0.9 1.160499907791320e-02 1.204501683071290e-02 1.185180808068810e-02

TABLE VII

THE ERROR OF CS-CS FOR p = 3 , v = 0.5

t Exact

Solution Mean solution of CS-CS Min error by CS-CS Max error by CS-CS Mean error by CS-CS

0.0 0.00000000 0.000000002488263 3.575699808794530e-10 7.543891859534880e-10 6.147292989429410e-09

0.1 0.00100000 0.001000004272664 1.726569095941020e-08 2.181228103091940e-07 7.953910263397570e-08

0.2 0.00800000 0.008000007493398 1.878053475928840e-08 3.178512749980880e-07 1.153399363086850e-07

0.3 0.02700000 0.027000008015559 9.453083996135980e-09 3.339483128535210e-07 1.241476972414320e-07

0.4 0.06400000 0.064000006680766 6.165678703706770e-09 3.004224534697290e-07 1.177731709646930e-07

0.5 0.12500000 0.125000004330639 2.352477077027790e-08 2.512822264294500e-07 1.062012284040270e-07

0.6 0.21600000 0.216000001806796 3.807320952953220e-08 2.205361613605290e-07 1.076849300165240e-07

0.7 0.34300000 0.342999999950857 4.526001246007990e-08 2.421927878248910e-07 1.310367756746090e-07

0.8 0.51200000 0.511999999604442 4.053419666583120e-08 3.502606354954810e-07 1.755128204328220e-07

0.9 0.72900000 0.729000001609168 1.934478000009680e-08 5.787482337815680e-07 2.677919390126070e-07

TABLE VIII

THE ERROR OF CS-CS FOR p = 3 , v = 1.5

t Exact Solution Mean solution of CS- CS Min error by CS-CS Max error by CS-CS Mean error by CS-CS

0.0 0.00000000 0.001000000000000 2.168404344971009e-19 6.505213034913030e-19 5.348730717595160e-19

0.1 0.00100000 0.008000000000000 0 3.469446951953610e-18 1.272130549049660e-18

0.2 0.00800000 0.027000000000000 0 6.938893903907230e-18 2.775557561562890e-18

0.3 0.02700000 0.064000000000000 0 1.387778780781450e-17 4.625929269271490e-18

0.4 0.06400000 0.125000000000000 0 1.387778780781450e-17 6.476300976980080e-18

0.5 0.12500000 0.216000000000000 0 2.775557561562890e-17 1.110223024625160e-17

0.6 0.21600000 0.343000000000000 0 5.551115123125780e-17 5.551115123125780e-18

0.7 0.34300000 0.512000000000000 0 0 0

0.8 0.51200000 0.729000000000000 0 1.110223024625160e-16 7.401486830834380e-18

0.9 0.72900000 1.000000000000000 0 1.110223024625160e-16 1.110223024625160e-17

REFERENCES

[1] C. Lubich, “Discretized fractional calculus,” SIAM Journal on

Mathematical Analysis, vol.17, no.3, pp. 704–719, 1986.

[2] N. J. Ford, J. A. Connolly , “Systems-based decomposition schemes

for the approximate solution of multi-term fractional differential

equations,” Journal of Computational and Applied Mathematics,

vol.229, no. 2, pp.382–391, 2009.

[3] I. Podlubny, Fractional differential equations (Book style), 1999.

[4] Z. M. Odibat, “Computational algorithms for computing the fractional

derivatives of functions,” Elsevier Science Publishers B. V, 2009.

[5] Z. Odibat, S. Momani, “Numerical methods for nonlinear partial

differential equations of fractional order,” Applied Mathematical

Modelling, vol.32, no.1, pp. 28–39, 2008.

[6] Z. Odibat, S. Momani, “Modified homotopy perturbation method:

application to quadratic differential equation of fractional order,”

Chaos, Solitons Fractal , vol.36, no.1, pp.167–174, 2008.

[7] Y. Tan, S. Abbasbandy, “Homotopy analysis method for quadratic

Riccati differential equation,” Communications in Nonlinear Science

and Numerical Simulation, vol.13, no.3, pp. 539–546, 2008.

[8] G. C. Wu, E. W. M. Lee , “Fractional variational iteration method and

its application,” Physics Letters A, vol.374, no.25, pp. 2506–2509,

2010.

[9] S. Abbasbandy, “A new application of He's variational iteration

method for quadratic Riccati differential equation by using Adomian's

polynomials,” Journal of Computational & Applied Mathematics,

vol.207, no.1, pp. 59–63, 2007.

[10] H. S. Yazdi, R. Pourreza, “Unsupervised adaptive neural-fuzzy

inference system for solving differential equations,” Applied Soft

Computing, vol.10, no.1, pp.267–275, 2010.

[11] A. Pedas, E. Tamme, “Spline collocation methods for linear multi-term

fractional differential equations,” Journal of Computational &

Applied Mathematics, vol.236, no.2, pp. 167–176, 2011.

[12] E.A.Rawashdeh, “Numerical solution of fractional integro–differential

equations by collocation method,” Applied Mathematics and

Computation, vol.176, no.1, pp.1–6, 2006.

[13] A. Saadatmandi, M. Dehghan, “A new operational matrix for solving

fractional-order differential equations,” Computers & Mathematics

with Applications, vol.59, no.3, pp. 1326–1336, 2010.

[14] M. M. Khader, “Numerical solution of nonlinear multi-order fractioanl

defferential equations by implementation of the operatioanl matrix of

fractional derivative,” Stud Nonlinear Sci , vol.2, no.1, pp. 5–12, 2011.



______________________________________________________________________________________

[15] F. Mohammadi, M. M. Hosseini, “A comparative study of numerical

methods for solving quadratic differential equations,” Journal of the

Franklin Institude, vol.348, no.8, pp. 1787–1796, 2011.

[16] M. L. Li, L. F. Wang, “Numerical solution of fractional partial

differential equation of parabolic type using Chebyshev wavelets

method,” Engineering Letters, vol. 26, no. 2, pp. 224–227, 2018.

[17] N. A. Khan, A. Ara, M. Jamil , “An efficient approach for solving the

Riccati equation with fractional orders,” Computers and Mathematics

with Applications, vol.61, no.9, pp. 2683–2689, 2011.

[18] M. A. Z. Raja, J. A. Khan, I. M. Qureshi, “Evolutionary computation

technique for solving Riccati differential equation of arbitrary order,”

World Academy of Science, vol.58, no.58, pp.531–536, 2009.

[19] M. A. Z. Raja, “A new stochastic approach for solution of differential

Riccati equation of fractional order,” Annals of Mathematics &

Artificial Intelligence, vol.60, no.3–4, pp. 229–250, 2010.

[20] M. A. Z. Raja, M. A. Manzar, R. Sama, “An efficient computational

intelligence approach for solving fractional order Riccati equations

using ANN and SQP,” Applied Mathematical Modelling, vol.39,

no.10–11, pp.3075–3093, 2015.

[21] X. S. Yang, S. Deb , “Engineering optimization by cuckoo search,”

International Journal of Mathematical Modelling and Numerical

Optimisation, vol.1, no.4, pp.330–343, 2010.

[22] X. S. Yang, “Cuckoo search for inverse problems and simulated-driven

shape optimization,” Journal of Computational Methods in Sciences

and Engineering, vol.12, no.1–2, pp. 129–137, 2012.

[23] V. Bhargava , S. E. K. Fateen, A. Bonillapetriciolet , “Cuckoo Search:

A new nature-inspired optimization method for phase equilibrium

calculations,” Fluid Phase Equilibria, vol.337, no.337, pp. 191–200,

2013.

[24] A. H. Gandomi, X. S. Yang, A. H. Alavi , “Cuckoo search algorithm: a

metaheuristic approach to solve structural optimization problems,”

Engineering With Computers, vol.29, no.1, pp.17–35, 2013.

[25] X. M. Zhang, “Parameter estimation of shallow wave equation via

cuckoo search,” Neural Computing & Applications, vol.28, no.12,

pp.4047–4059, 2017.

[26] X. M. Zhang, Q. Wan, Y. H. Fan, “Applying modified cuckoo search

algorithm for solving systems of nonlinear equations,” Neural

Computing and Applications, vol.31, no.2, pp.553–576, 2019.

[27] G. Chen, S. Qiu, Z. Zhang, Z. Sun, “Quasi-oppositional Cuckoo Search

Algorithm for Multi-objective Optimal Power Flow,” IAENG

International Journal of Computer Science, vol.45, no. 2, pp.

255–266, 2018.

[28] S. A. Medjahed, T. A. Saadi, A. Benyettou, M. Ouali, “Binary cuckoo

search algorithm for band selection in hyperspectral image

classification,” IAENG International Journal of Computer Science,

vol. 42, no.3, pp.183–191, 2015.

[29] X. Li, “Numerical solution of fractional differential equations using

cubic B-spline wavelet collocation method,” Communications in

Nonlinear Science & Numerical Simulation, vol.17, no.10, pp.

3934–3946, 2012.

[30] D. Kai, “An algorithm for the numerical solution of differential

equations of fractional order,” Electron. Trans. Numer. Anal, vol.5,

no.1, pp.1–6, 1997.

[31] X. Cai, J. H. Chen, “Numerical method for the fractional order ordinary

differential equation,” Journal of Jimei Uiversity (Natural Science),

vol.12, no.4, pp.367–370, 2007.



______________________________________________________________________________________

Applying Cuckoo Search Algorithm to Solve …II. M ATHEMATICAL M ODEL FOR SOLVING FRDE For positive real number v, 01d dn v n, the order Caputo fractional derivative of the function

Documents